A048399 -id:A048399 - cp's OEIS frontend

A048398 Primes with consecutive digits that differ exactly by 1.

2, 3, 5, 7, 23, 43, 67, 89, 101, 787, 4567, 12101, 12323, 12343, 32321, 32323, 34543, 54323, 56543, 56767, 76543, 78787, 78989, 210101, 212123, 234323, 234343, 432121, 432323, 432343, 434323, 454543, 456767, 654323, 654343, 678767, 678989

Offset: 1

Patrick De Geest, Apr 15 1999

Or, primes in A033075. - Zak Seidov, Feb 01 2011

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 67, p. 23, Ellipses, Paris 2008.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms n = 1..1500 from Zak Seidov)

Cf. A033075, A048399-A048405, A052016, A052017, A006055.

Cf. A010051; intersection of A033075 and A000040.

a048398 n = a048398_list !! (n-1)
a048398_list = filter ((== 1) . a010051') a033075_list
-- Reinhard Zumkeller, Feb 21 2012, Nov 04 2010
(Python 3.2 or higher)
from itertools import product, accumulate
from sympy import isprime
A048398_list = [2,3,5,7]
for l in range(1,17):
    for d in [1,3,7,9]:
        dlist = [d]*l
        for elist in product([-1,1],repeat=l):
            flist = [str(d+e) for d,e in zip(dlist,accumulate(elist)) if 0 <= d+e < 10]
            if len(flist) == l and flist[-1] != '0':
                n = 10*int(''.join(flist[::-1]))+d
                if isprime(n):
                    A048398_list.append(n)
A048398_list = sorted(A048398_list) # Chai Wah Wu, May 31 2017

Select[Prime[Range[10000]], # < 10 || Union[Abs[Differences[IntegerDigits[#]]]] == {1} &]

A048405 Primes with consecutive digits that differ exactly by 8.

Original entry on oeis.org

2, 3, 5, 7, 19, 191, 919, 919191919, 91919191919, 91919191919191919, 91919191919191919191919, 191919191919191919191919191919191

Offset: 1

Patrick De Geest, Apr 15 1999

The next term (a(13)) has 133 digits. - Harvey P. Dale, Jan 04 2023

			2 is a term since all its consecutive digits differ by 5 (there aren't any).
19 is a term because 1 and 9 differ by 8.
23 is not a term because its consecutive digits differ only by 1.

Sean A. Irvine, Table of n, a(n) for n = 1..13

Cf. A048398, A048399, A048400, A048401, A048402, A048403, A048404, A048410.

Module[{nn=500,nine,one},one=Select[Table[FromDigits[PadRight[{},n,{1,9}]],{n,nn}],PrimeQ];nine=Select[Table[FromDigits[PadRight[{},n,{9,1}]],{n,nn}],PrimeQ];Sort[Join[{2,3,5,7},nine,one]]] (* Harvey P. Dale, Jan 04 2023 *)

Offset corrected by Sean A. Irvine, Jun 16 2021

A048400 Primes with consecutive digits that differ exactly by 3.

Original entry on oeis.org

2, 3, 5, 7, 41, 47, 14741, 14747, 74747, 1414741, 1474141, 7414741, 4141414747, 4147414147, 14141414141, 14141414741, 14141474741, 14141474747, 14147414741, 14147474141, 14147474147, 14741414747, 74141414147, 74141414741, 74147414741, 74741474741, 74747414141

Offset: 1

Patrick De Geest, Apr 15 1999

All terms with more than a single digit must comprise only the digits 1, 4, and 7, because no number comprising the digits 2, 5, and 8 or the digits 3, 6, and 9 can be prime. - Harvey P. Dale, Mar 01 2023

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Cf. A033081, A048398, A048399, A048401, A048402, A048403, A048404, A048405.

Join[{2,3,5,7},Table[Select[FromDigits/@Tuples[{1,4,7},n],PrimeQ[#]&& Union[ Abs[ Differences[ IntegerDigits[ #]]]]=={3}&],{n,11}]//Flatten] (* Harvey P. Dale, Mar 01 2023 *)

More terms from Naohiro Nomoto, Jul 28 2001

More terms from Sean A. Irvine, Jun 15 2021

A048401 Primes with consecutive digits that differ exactly by 4.

Original entry on oeis.org

2, 3, 5, 7, 37, 59, 73, 151, 373, 15959, 95959, 515951, 595159, 595951, 9515959, 51515159, 159595151, 159595951, 5151515951, 5159515159, 5159515951, 5951515151, 5951515951, 5959515151, 5959595951, 15151595951, 15951515159

Offset: 1

Patrick De Geest, Apr 15 1999

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Cf. A048398, A048399, A048400, A048402, A048403, A048404, A048405, A048406.

pd[{a_,b_,c___}]:=Flatten[Table[Select[FromDigits/@Select[Tuples[ {a,b,c},n],Union[Abs[Differences[#]]]=={4}&],PrimeQ],{n,11}]]; Union[Join[{2,3,5,7},pd[{1,5,9}],pd[{3,7}]]] (* Harvey P. Dale, Aug 23 2011 *)

More terms from Naohiro Nomoto, Jul 28 2001

A048402 Primes with consecutive digits that differ exactly by 5.

Original entry on oeis.org

2, 3, 5, 7, 61, 83, 383, 727, 72727, 94949, 1616161, 383838383, 727272727, 383838383838383, 38383838383838383, 72727272727272727, 94949494949494949, 383838383838383838383

Offset: 1

Patrick De Geest, Apr 15 1999

Harvey P. Dale, Table of n, a(n) for n = 1..30

Cf. A048398, A048399, A048400, A048401, A048403, A048404, A048405, A048407.

Module[{nn=50,w1,w2},w1=Flatten[Table[Select[FromDigits/@Table[ PadRight[ {},n,{a,a+5}],{n,2,nn}],PrimeQ],{a,4}]];w2=Flatten[Table[Select[ FromDigits/@ Table[PadRight[{},n,{a+5,a}],{n,2,nn}],PrimeQ],{a,4}]];Join[ {2,3,5,7},w1,w2]//Union] (* Harvey P. Dale, Jan 09 2021 *)

A048404 Primes with consecutive digits that differ exactly by 7.

Original entry on oeis.org

2, 3, 5, 7, 29, 181, 929, 18181, 929292929, 18181818181818181818181818181818181818181818181818181818181818181818181818181

Offset: 1

Patrick De Geest, Apr 15 1999

The next term (a(11)) has 163 digits. - Harvey P. Dale, Mar 23 2023

Sean A. Irvine, Table of n, a(n) for n = 1..13

Cf. A048398, A048399, A048400, A048401, A048402, A048403, A048405, A048409.

Module[{s18,s81,s29,s92},s18=Select[Table[FromDigits[PadRight[{},n,{1,8}]],{n,1,181,2}],PrimeQ]; s81=Select[Table[FromDigits[PadRight[{},n,{8,1}]],{n,2,182,2}],PrimeQ];s29 = Select[ Table[FromDigits[PadRight[{},n,{2,9}]],{n,2,182,2}],PrimeQ]; s92 =Select[Table[ FromDigits[ PadRight[{},n,{9,2}]],{n,1,183,2}],PrimeQ]; Join[{2,3,5,7},s18,s81,s29,s92]//Sort] (* Harvey P. Dale, Mar 23 2023 *)

A048403 Primes with consecutive digits that differ exactly by 6.

Original entry on oeis.org

2, 3, 5, 7, 17, 71, 1717171717171717171717171717171, 1717171717171717171717171717171717171

Offset: 1

Patrick De Geest, Apr 15 1999

From Andrew Howroyd, Aug 13 2024: (Start)
Terms with more than 1 digit have digits alternating between 1 and 7.
No more terms < 10^3000. (End)

Primes in A048408.

Cf. A048398, A048399, A048400, A048401, A048402, A048404, A048405.

upto(limit)={my(L=List([t|t<-[2,3,5],t<=limit]),m=1); while(mAndrew Howroyd, Aug 13 2024

Offset changed by Andrew Howroyd, Aug 13 2024

A089316 Prime worms [successive digit differences with absolute value of 2].

Original entry on oeis.org

131, 313, 353, 757, 797, 35353, 35753, 75797, 79757, 97579, 3131353, 3135313, 3531313, 7535797, 313131353, 313135313, 313579753, 353535313, 357531313, 357531353, 357535753, 357575753, 357975353, 753535357, 757975357, 975353579

Offset: 0

Enoch Haga, Dec 25 2003

			a(4)=797; first and last digits are 7; abs(7-9)=2; abs(9-7)=2; the worm is 7.

Carlos Rivera, Puzzle 246. The worm, The Prime Puzzles and Problems Connection.

Cf. A089291.

This is a subset of A048399. Cf. A089291, A089315, A089317, A048398-A048405.

pwQ[n_]:=Module[{idn=IntegerDigits[n]},First[idn]==Last[idn]&&Union[Abs[ Differences[idn]]]=={2}]; Select[Prime[Range[50000000]],pwQ] (* Harvey P. Dale, Mar 26 2013 *)

Select prime numbers having the same first and last digits; if the uniform absolute value of successive digit differences is 2, add to sequence.

A242846 Palindromic primes of the form ababa...aba containing only the digits 1 and 3.

Original entry on oeis.org

131, 313, 1313131313131313131313131, 313131313131313131313131313131313131313131313131313

Offset: 1

Felix Fröhlich, May 23 2014

The next two terms both start with 3 and have 83 and 225 digits, respectively. Those are the only other terms with fewer than 352 digits. Cf. A062216.

Cf. A062216, A032758, A048399, A235154.

Module[{nn=60,a,b},a=Table[FromDigits[Join[PadRight[{},2n,{1,3}],{1}]],{n,nn}];b=Table[FromDigits[Join[PadRight[{},2n,{3,1}],{3}]],{n,nn}];Select[Sort[Join[a,b]],PrimeQ]] (* Harvey P. Dale, Sep 07 2020 *)

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A048398 Primes with consecutive digits that differ exactly by 1.

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A048405 Primes with consecutive digits that differ exactly by 8.

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A048400 Primes with consecutive digits that differ exactly by 3.

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A048401 Primes with consecutive digits that differ exactly by 4.

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A048402 Primes with consecutive digits that differ exactly by 5.

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A048404 Primes with consecutive digits that differ exactly by 7.

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A048403 Primes with consecutive digits that differ exactly by 6.

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A089316 Prime worms [successive digit differences with absolute value of 2].

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A242846 Palindromic primes of the form ababa...aba containing only the digits 1 and 3.

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